In your examples, "a number/variety/lot of..." can be replaced with other synonyms such as "several" or "multiple". I'd look at the word which follows "of", and in all cases you have plural nouns, so use "are".
The small issue here is that we often use "there's" in speech, even when the subject is plural. I mean the contracted form, not the full form. Maybe this will become widely accepted decades from now.